ECEn 370 Midterm Name:

RED- You can write on this exam.

ECEn 370

Introduction to ProbabilitySection 001

Midterm

Winter, 2011

Instructor

Professor Brian Mazzeo

Closed Book - You can bring one 8.5 X 11 sheet of handwritten notes on both sides.

Graphing or Scienti�c Calculator Allowed

3-Hour Suggested Time Limit

IMPORTANT!

• WRITE YOUR NAME on every page of the exam.

• Answer questions 1-26 on the provided bubble sheet.

• Questions 1-26 are worth 1 point each.

• Do not discuss the exam with other students.

• NOTE: Use all of the digits on your calculator, or fractions, before at the end rounding to the number ofsigni�cant digits used in the problem.

1

ECEn 370 Midterm Name:

1. Let A and B be two independent events in S. It is known that P (A ∩B) = 0.09 and P (A ∪B) = 0.51.Calculate P (A).

A) 0.160B) 0.200C) 0.250D) 0.300E) 0.350F) 0.400G) 0.450H) 0.500I) 0.640J) None of the Above

2. Out of the students in a class, 50% are tall, 30% are athletes, and 20% fall into both categories. Deter-mine the probability that a randomly selected student in this class is neither tall nor an athlete.

A) 10%B) 20%C) 30%D) 40%E) 50%F) 60%G) 70%H) 80%I) 90%J) None of the Above

3. A batch of forty items is inspected by testing four randomly selected items. If one of the four is defective,the batch is rejected. What is the probability that the batch is accepted if it contains �ve defectives?

A) 0.008B) 0.125C) 0.381D) 0.491E) 0.493F) 0.500G) 0.573H) 0.647I) 0.812J) None of the Above

4. Sixty students, including Joe and Jane, are to be split into three classes of equal size, and this is to be done atrandom. What is the probability that Joe and Jane end up in the same class?

A) 0.106B) 0.107C) 0.317D) 0.322E) 0.328F) 0.333G) 0.339H) 0.966I) 0.983J) None of the Above

2

ECEn 370 Midterm Name:

5. A power utility can supply electricity to a city from 10 di�erent power plants (not very good ones). Each powerplant fails with probability 0.6. Suppose that two power plants are necessary to keep the city from a black-out.Find the probability that the city will experience a black-out.

A) 0.004B) 0.006C) 0.010D) 0.040E) 0.046F) 0.060G) 0.160H) 0.400I) 0.540J) None of the Above

6. A number source generates digits 2, 1, and 0. The probability of a 2 is 0.3, of a 1 is 0.6, and of a 0 is 0.1. A�ve-digit sequence is generated. What is the probability that two 1s will occur in the sequence?

A) 0.100B) 0.150C) 0.230D) 0.250E) 0.300F) 0.317G) 0.600H) 0.683I) 0.700J) None of the Above.

7. A random variable is called a Rayleigh random variable if its pdf is given by

fX(x) =

{xσ2 e

−x2/(2σ2) x > 00 x < 0

The radial distance [in meters (m)] of the landing point of a parachuting sky diver from the center of a target areais known to be a Rayleigh random variable with parameter σ2. If the probability that the sky diver will land withina radius of 10 m from the center of the target area is 0.1, what is σ2?

A) 0 < σ2 < 100B) 100 ≤ σ2 < 200C) 200 ≤ σ2 < 300D) 300 ≤ σ2 < 400E) 400 ≤ σ2 < 500F) 500 ≤ σ2 < 600G) 600 ≤ σ2 < 700H) 700 ≤ σ2 < 800I) 800 ≤ σ2J) None of the Above.

3

ECEn 370 Midterm Name:

8. If at �rst you don't succeed, try, try, try again. A computer will successfully send a message across a networkwith probability 0.6. The computer will retry sending the message until it is successfully sent. Given that we knowthat the computer will successfully transmit the message on or before the third attempt, what is the probabilitythat the computer successfully sends the message on the second attempt?

A) 0.124B) 0.240C) 0.326D) 0.375E) 0.600F) 0.625G) 0.775H) 0.938I) 0.946J) None of the Above

9. Suppose that this year the probability of at least one date is 0.9, the probability of at least one missinghomework is 0.6, and the probability of at least one bombed test is 0.2. What is the probability that this year youwill have at least one date or at least one missing homework or at least one bombed test?

A) 0.032B) 0.108C) 0.324D) 0.648E) 0.712F) 0.892G) 0.912H) 0.968I) 0.983J) None of the Above

10. A random variable X has a probability density function given by

fX(x) =

{(x−3)3

4 , if 3 ≤ x ≤ 5,0, otherwise.

Find the cumulative distribution function, FX(x), and compute the value of the following expression:

FX(1) + FX(4) + FX(9)

A) 0B) 1/16C) 1/4D) 17/16E) 2F) 33/16G) 4H) 54I) 1297/16J) None of the Above

4

ECEn 370 Midterm Name:

11. You have the following joint PMF of random variables X and Y :

x

y1/10

3/10

3/10

3/10

0 0 0 0

00

0

000

0

0

0

0

0

0

1 2 3 4 5

1

2

3

4

Find E[Z] where Z = XY

A) 0.58B) 0.96C) 2.20D) 2.90E) 6.38F) 7.40G) 7.52H) 10.9I) 24.9J) None of the Above

12. You have three ten-sided dice numbered from 1 to 10. Each face has an equal probabilitiy of appearingduring a roll (uniformly distributed). The outcomes, representing each die, are the random variables X, Y , and Z.Find P (max(X,Y, Z) = 4).

A) 0.005B) 0.016C) 0.027D) 0.037E) 0.064F) 0.091G) 0.125H) 0.216I) 0.400J) None of the Above

13. Suppose that X, Y , and Z are independent random variables. E[X] = 4, E[Y ] = 2, and E[Z] = 1. var(X) = 3,var(Y ) = 2, and var(Z) = 1. Compute the following expressions.E[3XY Z − Y ] + var(X − Y ) + E[XY 2]− var(−3Z)

A) 18B) 30C) 32D) 34E) 38F) 42G) 51H) 52I) 60J) None of the Above

5

ECEn 370 Midterm Name:

14. The new fad of Greenie Babies started. There are �ve plush animals. Stores of McTacoKing will give you oneof the �ve with each meal purchase, but you don't know which one you will get. What is the expected number, n,of McTacoKing meal purchases you need to make to collect them all?

A) 0 < n ≤ 5B) 5 < n ≤ 7C) 7 < n ≤ 9D) 9 < n ≤ 11E) 11 < n ≤ 13F) 13 < n ≤ 15G) 15 < n ≤ 17H) 17 < n ≤ 19I) 19 < n ≤ 21J) None of the Above

15. You have the following joint PDF with a constant density, fX,Y (x, y) = c, in the shaded region and is zerooutside that region:

321

1

y

xFind E[X] + E[Y ]

A) 7/6B) 4/3C) 3/2D) 5/3E) 11/6F) 2G) 13/6H) 7/3I) 5/2J) None of the Above

16. A binary signal S is transmitted, and we are given that P (S = 1) = 0.6 and P (S = −1) = 0.4. The receivedsignal is Y = N + S, where N is normal noise, with zero mean and unit variance, independent of S. What is theprobability that S = 1, as a function of the observed value 0.7 of Y ?

A) 0.141B) 0.167C) 0.250D) 0.270E) 0.677F) 0.700G) 0.730H) 0.802I) 0.859J) None of the Above

6

ECEn 370 Midterm Name:

17. The time until a small meteorite �rst lands anywhere in the Sahara desert is modeled as an exponential randomvariable with a mean of 10 days. The time is currently midnight. What is the probability that a meteorite �rstlands some time between 6 a.m. and 6 p.m. of the �rst day?

A) 0.005B) 0.025C) 0.048D) 0.072E) 0.081F) 0.082G) 0.093H) 0.097I) 0.190J) None of the Above

18. We are told that the joint PDF of the random variables X and Y is a constant c on the set S shown in the�gure below and is zero outside. We wish to determine the value of c and the marginal PDFs of X and Y .

321

1

y

x

2

3

4

S

Compute the following:c+ fY (2.5) + fY (1.5) + fX(0.5) + fX(1.5) + fX(2.5)

A) 2B) 9/4C) 5/2D) 11/4E) 3F) 13/4G) 7/2H) 15/4I) 4J) None of the Above

19. From the �gure above in problem 18, compute the following where FX,Y is the CDF of the joint PDF.FX,Y (−1,−1) + FX,Y (1.5, 1.5) + FX,Y (3.5, 1.5) + FX,Y (2, 2) + FX,Y (2.5, 4.5) + FX,Y (4.5, 6)

A) 29/16B) 30/16C) 31/16D) 32/16E) 33/16F) 34/16G) 35/16H) 36/16I) 37/16J) None of the Above

7

ECEn 370 Midterm Name:

20. Suppose that the random variable X has the piecewise constant PDF

fX(x) =

1/3, if 0 ≤ x ≤ 1,2/3, if 1 < x ≤ 20, otherwise

Evaluate the following: E[X] + var(X)

A) 5/3B) 11/6C) 2D) 13/6E) 7/3F) 5/2G) 8/3H) 17/6I) 3J) None of the Above

21. Each morning, Hungry Hungry Horace eats some sausages. On any given morning, the number of sausages heeats is equally likely to be 1, 2, 3, 8, or 9, independent of what he has done in the past. Let X be the number ofsausages that Harry eats in 10 days. Compute q = E[X] + var(X).

A) 0 < q ≤ 10B) 10 < q ≤ 30C) 30 < q ≤ 50D) 50 < q ≤ 70E) 70 < q ≤ 90F) 90 < q ≤ 110G) 110 < q ≤ 130H) 130 < q ≤ 150I) 150 < q ≤ 170J) None of the Above

22. A stock market trader buys 1 share of stock A and 2 shares of stock B. Let X and Y be the price changes of Aand B, respectively, over a certain time period, and assume that the joint PMF of X and Y is uniform over the setof integers x and y satisfying

−1 ≤ x ≤ 2, 0 ≤ y − x ≤ 1

Find the mean of the trader's pro�ts (or losses if negative).

A) -1B) -1/2C) 0D) 1/2E) 1F) 3/2G) 2H) 5/2I) 3J) None of the Above

8

ECEn 370 Midterm Name:

23. Let X be a random variable with PMF

pX(x) =

{x2/a, if x = −3,−2,−1, 0, 1, 2, 3,0, otherwise

Find a+ E[X] + var(X)

A) 17B) 19C) 29D) 34E) 35F) 36G) 76H) 77I) 78J) None of the Above

24. Let X be a random variable that takes values from 0 to 9 with equal probability 1/10. Find the PMF of therandom variable Y = X mod(3).

Calculate pY (0) + pY (1)

A) 1/10B) 2/10C) 3/10D) 4/10E) 5/10F) 6/10G) 7/10H) 8/10I) 9/10J) None of the Above

25. Al thows darts at a circular target of radius 1 m and is equally likely to hit any point on the target. LetX be the distance of Al's hit from the center. Find E[X] + var(X).

A) 0.896B) 0.898C) 0.900D) 0.902E) 0.904F) 0.906G) 0.908H) 0.910I) 0.912J) None of the Above

9

ECEn 370 Midterm Name:

26. The scores on an exam are normally distributed with a mean of 80 and a standard deviation of 10. If you knowyour score, x, is above 75% of your peers (but 25% of your peers are above you), then choose the correct rangewhere your score falls.

NOTE: Values of the standard normal CDF are Φ(0) = 0.500, Φ(0.2) = 0.5793, Φ(0.4) = 0.6554, Φ(0.6) = 0.7257,Φ(0.8) = 0.7881, Φ(1.0) = 0.8413

A) 0 < x < 65B) 65 < x < 70C) 70 < x < 75D) 75 < x < 80E) 80 < x < 85F) 85 < x < 90G) 90 < x < 95H) 95 < x < 100I) x > 100J) None of the Above

10